As Argentina's maverick libertarian President Javier Milei marks one year in office, his efforts to revive the economy are still a work in progress - but his policies are proving influential in the US.

Milei came to power with a mission to cut state spending in a country that had been living beyond its means for years.

Despite his tough austerity measures and a continued rise in poverty rates, he is still supported by just over half the population, according to a survey carried out earlier this month by the CB Consultora organisation.

That level of popularity is similar to that of Donald Trump right now. Roughly half of US voters backed the president-elect in last month's presidential contest - and Trump has hailed Milei as a man who can "make Argentina great again".

Meanwhile, tech billionaire Elon Musk, who looks set to play a key role in the incoming US administration, has also praised Milei, saying Argentina is "experiencing a giant improvement" under his leadership.

But what is it that Trump and Musk see in Milei? And are they as close ideologically as is often assumed?

Milei's biggest achievement so far, the one which is most prized by Argentines, is his success in cutting inflation. But he has caused a stir in the US because of his deregulation drive, which has been seized on by small-government activists keen to shrink the size of the state in Washington along the lines of what is happening in Buenos Aires.

In Milei's initial package of measures, he slashed state subsidies for fuel and cut the number of government ministries by half.

Now he is trying to force through plans for a mass sell-off of state-run companies, including the country's flagship airline Aerolineas Argentinas, which has already been privatised once before being renationalised in 2008.

All this is music to the ears of Elon Musk, who is being tasked with similar cost-cutting initiatives under the banner of the so-called Department of Government Efficiency - a misleading name, since it is an advisory body, not an official government department.

Musk and his co-leader in the department, fellow billionaire Vivek Ramaswamy, have said they want to slash federal regulations, oversee mass layoffs and shut down some agencies entirely.

Musk has spoken of cutting federal government spending by $2tn (£1.6tn) - about one-third of annual expenditure. According to him, Milei is doing "a fantastic job" in Argentina by "deleting entire departments" - and he would like to follow suit in the US, with Trump's blessing.

But long-time Latin America observers are sceptical.

Monica de Bolle, senior fellow at the Peterson Institute for International Economics in Washington, says that "taking inspiration from Milei to reduce the size of government doesn't make any sense".

"The situation in Argentina is very particular to Argentina, because it was about the removal of decades of mismanagement of public resources. That has nothing to do with the US."

Ms de Bolle says Argentina had no choice but to take action, because government overspending was so excessive that the country was "bursting into crisis every few years".

"That is appropriate for Argentina, but for nobody else."

Marcelo J García, Buenos Aires-based director for the Americas at global consulting firm Horizon Engage, says Milei's decision to wield a chainsaw on the campaign trail as a sign of his approach to government was a "masterpiece" of political marketing that has "captured the imagination of small-state activists across the globe".

But he argues that while Musk's own business interests would benefit from less government regulation, that's not necessarily what Trump wants.

"I'm not sure that the Trump platform is compatible with a Milei-type chainsaw small government," he told the BBC.

He points out that Trump's policies "require big government in some areas", such as the building of border walls and mass deportations of illegal immigrants. "You can't do those kinds of massive programmes with small government."

In Milei's view, infrastructure projects are best left to the private sector and have nothing to do with government.

Milei and Trump are on the same side in the global culture wars, denouncing what they see as the "woke agenda". But in economic terms, their ideas are very different.

Milei is a passionate free-trader, and Argentina is a member of the South American trading bloc Mercosur, which also includes Brazil, Paraguay, and Uruguay.

While he is in favour of Mercosur's recent free-trade deal with the European Union, he doesn't like the way that the organisation refuses to let its individual member countries strike their own deals. As a result, he says Mercosur "has ended up becoming a prison".

"If the bloc is not a dynamic engine that facilitates trade, boosts investment and improves the quality of life of all the citizens of our region, what is the point of it?" he said at the Mercosur summit in Uruguay earlier this month where the deal with the EU was signed.

Trump also has beef with his own regional trade alliance, the US-Mexico-Canada Agreement (USMCA), but for reasons that are the opposite of Milei's.

Trump wants to renegotiate the USMCA, a deal that he himself put together during his first term in office, as a way of protecting US manufacturing and safeguarding US jobs.

He has even found a way of weaponising the alliance by threatening to impose a blanket 25% tariff on goods from both Canada and Mexico unless they secure their shared borders with the US.

Monica de Bolle doubts that Trump shares Musk's enthusiasm for a smaller state: "You can't be a populist nationalist and care about the size of government. So Trump doesn't care. He put Elon there because it's kind of fun to have someone there making noise."

The economic debate is set to run and run, in both the US and Argentina. But ultimately, if one half of your population supports you, it means the other half doesn't. Trump will have to deal with that after his inauguration on 20 January, but Milei is already having to cope with his own polarised population.

As Marcelo J García sees it, Milei is a "divisive leader" who has made no attempt to win over his opponents.

"The other half of the country that did not support him will arguably never support him, no matter how well the economy does, because he doesn't want them to support him," he says.

"Leaders tend to want to be liked by everyone. That's not the case with Milei," he adds.

In his view, this is a real weakness: "You don't build a long-term sustainable political project if you don't move towards the people who didn't vote for you."

Milei's next big test of public opinion will come in October 2025, when Argentina holds midterm elections. That could prove crucial in deciding whether his small-government revolution determines the country's future - or whether, like previous attempts at reform, it runs out of steam.